1. Normal Probability Distributions
Chapter 5
Normal Probability Distributions
Larson/Farber 4th ed

2. Chapter Outline
5.1 Introduction to Normal Distributions and the Standard Normal Distribution
5.2 Normal Distributions: Finding Probabilities
5.3 Normal Distributions: Finding Values
5.4 Sampling Distributions and the Central Limit Theorem
5.5 Normal Approximations to Binomial Distributions
Larson/Farber 4th ed

3. Normal Distributions: Finding Probabilities
Section 5.2
Normal Distributions: Finding Probabilities
Larson/Farber 4th ed

4. Section 5.2 Objectives
Find probabilities for normally distributed variables
Larson/Farber 4th ed

5. Probability and Normal Distributions
If a random variable x is normally distributed, you can find the probability that x will fall in a given interval by calculating the area under the normal curve for that interval.
μ = 500
σ = 100
600
μ =500 x
P(x < 600) = Area
Larson/Farber 4th ed

6. Probability and Normal Distributions
Standard Normal Distribution
600
μ =500
P(x < 600)
μ = σ = 100 x 1
μ = 0
μ = 0 σ = 1 z
P(z < 1)
Same Area
P(x < 500) = P(z < 1)
Larson/Farber 4th ed

7. Example: Finding Probabilities for Normal Distributions
A survey indicates that people use their computers an average of 2.4 years before upgrading to a new machine. The standard deviation is 0.5 year. A computer owner is selected at random. Find the probability that he or she will use it for fewer than 2 years before upgrading. Assume that the variable x is normally distributed.
Larson/Farber 4th ed

8. Solution: Finding Probabilities for Normal Distributions
Standard Normal Distribution
-0.80
μ = 0 σ = 1 z
P(z < -0.80) 2
2.4
P(x < 2)
μ = σ = 0.5 x
0.2119
P(x < 2) = P(z < -0.80) =
Larson/Farber 4th ed

9. Example: Finding Probabilities for Normal Distributions
A survey indicates that for each trip to the supermarket, a shopper spends an average of 45 minutes with a standard deviation of 12 minutes in the store. The length of time spent in the store is normally distributed and is represented by the variable x. A shopper enters the store. Find the probability that the shopper will be in the store for between 24 and 54 minutes.
Larson/Farber 4th ed

10. Solution: Finding Probabilities for Normal Distributions
-1.75 z
Standard Normal Distribution μ = 0 σ = 1
P(-1.75 < z < 0.75)
0.75 24 45
P(24 < x < 54) x
0.7734
0.0401 54
P(24 < x < 54) = P(-1.75 < z < 0.75)
= – =
Larson/Farber 4th ed

11. Example: Finding Probabilities for Normal Distributions
Find the probability that the shopper will be in the store more than 39 minutes. (Recall μ = 45 minutes and σ = 12 minutes)
Larson/Farber 4th ed

12. Solution: Finding Probabilities for Normal Distributions
Standard Normal Distribution μ = 0 σ = 1
P(z > -0.50) z
-0.50 39 45
P(x > 39) x
0.3085
P(x > 39) = P(z > -0.50) = 1– =
Larson/Farber 4th ed

13. Example: Finding Probabilities for Normal Distributions
If 200 shoppers enter the store, how many shoppers would you expect to be in the store more than 39 minutes?
Solution:
Recall P(x > 39) =
200(0.6915) =138.3 (or about 138) shoppers
Larson/Farber 4th ed

14. Example: Using Technology to find Normal Probabilities
Assume that cholesterol levels of men in the United States are normally distributed, with a mean of 215 milligrams per deciliter and a standard deviation of 25 milligrams per deciliter. You randomly select a man from the United States. What is the probability that his cholesterol level is less than 175? Use a technology tool to find the probability.
Larson/Farber 4th ed

15. Solution: Using Technology to find Normal Probabilities
Must specify the mean, standard deviation, and the x-value(s) that determine the interval.
Larson/Farber 4th ed

16. Section 5.2 Summary
Found probabilities for normally distributed variables
Larson/Farber 4th ed